Steklov Spectra and Div-curl Analysis

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This project centers on the analysis of div-curl systems of equations and on the solution of partial differential equations with physical boundary conditions in 3 dimensional regions. Such systems arise as mathematical models of wide classes of problems in continuum mechanics and electromagnetic field theories. Particular attention will be paid to describing solvability conditions and the approximation and representation of weak solutions of these systems. Some of the novelty involves the careful choice of scalar and vector potentials for these problems and the use of Steklov eigenfunctions to describe the effects of boundary data. Another class of problems is the development of variational principles that characterize solutions of initial value problems for evolution equations. The analysis performed under this award should help in the development of computational simulations and the numerical approximation of solutions to many important equations arising in science and engineering.